CURVE LINES. 



city, we may suppose the planes at right angles to one 

 another ; then, if they be represented by YAX, XAZ, 

 YAZ, (Fig. 14'.) and if it be known that a point M is 

 'late placed at the distance MM' from the first, MM" from 



jP** the second, and MM'" from the third, it follows, from 

 the property of parallel planes being every where 

 equally distant from one another, that if three planes 

 M'"MM", M"'MM', M"MM', be drawn parallel to 

 the former, at the given distances from them, the point 

 M will be found at their mutual intersection. 



39. The rectangular planes YAX, YAZ, XAZ, to 

 which the points of space are referred, are called the 

 Co-ordinate planes. They cut each other, two and two, 

 in the directions of three straight lines, AX, AY, AZ, 

 which pass through the same point A, and are perpen- 

 dicular to one another. 



40. From the nature of parallel planes, the distance 

 MM' may be measured on the line AZ, and is equal to 

 AR ; in like manner, the distance MM" may be mea- 

 sured on the line AY, and is equal to AQ ; and, lastly, 

 the distance MM'" may be measured on AX, and is 

 equal to AP. 



41. The straight lines AZ, AY, AX, upon which the 

 distances of the point M from the planes are reckoned, 

 are called the axes of the co-ordinates, and the point A 

 is called the origin. The line AP=MM'", the dis- 

 tance of M from the plane, which is perpendicular to 

 AX, may be denoted by x; and similarly, the line 

 AQ=MM", the distance of M from the plane perpen- 

 dicular to AY, may be expressed by y ; and the line 

 ARrrMM', the distance of M from the plane perpen- 

 dicular to AZ, by z. 



If, therefore, the three distances, AP, AQ, AR, are 

 found to be a, b, c, we have to determine the position 

 of the point M, these tlu-ee equations, 



x= a, y=b, z=cj 

 and as they suffice for that purpose, they may be called 

 the equations of the point M. 



The positions of the points M', M", M'", which are 

 called the projections of the point M on the tliree co- 

 ordinate planes, are determined by these equations, for 

 we have y=z b, x=a, for the co-ordinates of the point 

 M', the projection of M upon the plane YAX ; also, 

 x=a, z=c, for the co-ordinates of M", the projection 

 of the point M on the plane XAZ ; and z=c, y=b, 

 for the co-ordinates of M'", the projection of the point 

 M on the plane Z AY. From the nature of these equa- 

 tions, it is evident that any two of them being known, 

 the third is also known. 



42. It follows from what has been said, that all the 

 points of space being referred to three planes perpen- 

 dicular to one another, the points of each plane may be 

 naturally referred to two straight lines perpendicular to 

 one another, which are the intersections of that plane 

 with the two others. Thus, if each plane be denoted 

 by the co-ordinates which belong to it, the plane YAX 

 shall be that of x and y; the plane XAZ that of x and 

 z; and the plane ZAY that of y and z. 



43. Whatever has been said (Art. 9 11.) respecting 



the signs of the co-ordinates of plane curves, applies 

 equally here to the axes AX, AY, AZ ; and it follows 

 that the signs of the co-ordinates, x, y, z, shew the po- 

 sition of every point whatever in respect of the co-or- 

 dinate planes, it being understood that these are inde- 

 finitely extended, and that the straight lines AX, AY, 

 AZ, are each produced indefinitely both ways from A, 

 the common origin. 



Agreeably to this conventional mode of representing 

 the position o£ a point in space, according as it is in one 

 or other of the eight angles which have their common ver- 

 tex at A, the signs of its co-ordinates will be as follows : 



527 



+- v > +!/> + 2 > in the angle AXYZ, 

 + - x > +*/> — z > m the angle AXYz, 

 + - r > — y, +z, in the angle AXZy, 

 — x > +!/> + z > m the angle AYZ x, 

 +*> — V> — z , in the angle AXy z, 

 — x, -f y, — z, in the angle AY x z, 

 — x, — y, -f-z, in the angle AZxy, 

 —x, — y, — z, in the angle A xy z. 



44. Let M be any point in a surface of a known nature, 

 and which is referred to the three planes AM', AM", 

 AM"' ; We may suppose MM'"=j and MM"=^, its 

 distances from two of the planes, to have any magni- 

 tudes that admit of the point M being on the surface, 

 and corresponding to these, MM'zzz, its distance from 

 the third plane, will have a magnitude, depending on 

 the nature of the surface ; therefore the value of z de- 

 pends jointly on the values of x and y, so that these 

 being known, z is also known. 



The equation which expresses the relation of z to x 

 and y is called the equation of the surface. : so that like 

 as the nature of a line on a plane may be expressed by 

 an equation involving two indeterminate quantities, 

 the nature of a surface may, in general, be expressed by 

 an equation containing three indeterminate quantities. 



45. The position of a point M being given in space by 

 its co-ordinates x, y, z, its distance MA from A the ori- 

 gin is known. For we have AM 2 =AM' 2 - r .M'M 2 = 

 A?* + FM'z+M'M*=x* + i,* + z\ Hence also the co- 

 ordinates of any two points being given, their distance 

 is known ; for let x' ', ;/, z', be the co-ordinates of the 

 second point, and let us suppose the co-ordinate planes 

 to be transferred parallel to themselves, so that the ori- 

 gin may be at the first point; then the new co-ordinates 

 will be x' — x, y'—y, z' — ~, and the distance of the se- 

 cond point from the new origin, that is from the first 

 point, will be 



V |V-*) 2 + (y'-y) 2 + (z'-zy j 



46. The equation of a plane may be readily found from 

 the analytical expression for the distance between two 

 points, by considering it as a surface, every point of 

 which is equally distant from two given points. Let 

 the co-ordinates of the two points be 



a, b, c; a', b', c' ; 

 and let x, y, z, be the co-ordinates of any point in the 

 plane. The general expression for the distance of any 

 point in the plane, from the point whose co-ordinates 

 are a, b, c, will be 



V {O-«) 2 +(y-6) 2 +(*-<0 2 } ; 



and its distance from the point whose co-ordinates are 

 a', b', c', will in like manner be 



V { (*— «')* + (y— 1>7 + 0-c') 2 ) • 



By putting the squares of these distances equal to 

 one another, we get 



— 2 ax — 2 by — 2 cz + a^ + fr + c* 

 = — 2 a'x — 2 b'y — 2 c'z -|- a' 2 -f 6' 2 -4-c' 2 . 

 Let A=2 (a'— a), B=2 (b'—b), C = 2 (c'—c), — D 

 =a'2— a 2 -f-6' 2 — b'+c'* — c 2 , and we get 



Ax+Bv-r-Cz-f-DzrO 

 for the equation of a plane, from which it appears that the 

 equation of a plane is the most general that can be form- 

 ed by three indeterminate quantities of the first degree. 

 If we suppose one of the points to be at the origin of 

 the co-ordinates, so that a'=:0, i'=0, c'=0, the equa- 

 tion becomes 



2 (ax+by-{-cz)=a* + 6 2 + c 2 . 

 The point whose co-ordinates are a, b, c, is now in 

 a line drawn from the origin perpendicular to the plane, 

 and as far distant from it on one side as the origin is 



Curve 

 Surfaces. 



